{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "15e23b3b",
   "metadata": {},
   "source": [
    "# 高中物理热学公式大全\n",
    "\n",
    "## 1. 分子动理论\n",
    "\n",
    "### 分子直径数量级\n",
    "$$ d \\approx 10^{-10} {m} $$\n",
    "\n",
    "### 分子质量\n",
    "$$ m_0 = \\frac{M}{N_A} $$\n",
    "\n",
    "### 阿伏伽德罗常数\n",
    "$$ N_A = 6.02 \\times 10^{23} {mol}^{-1} $$\n",
    "\n",
    "### 分子数\n",
    "$$ N = nN_A = \\frac{m}{M}N_A $$\n",
    "\n",
    "## 2. 气体状态方程\n",
    "\n",
    "### 理想气体状态方程\n",
    "$$ pV = nRT = \\frac{m}{M}RT $$\n",
    "\n",
    "### 气体常数\n",
    "$$ R = 8.31 {J/(mol·K)} $$\n",
    "\n",
    "### 克拉珀龙方程\n",
    "$$ pV = \\nu RT $$\n",
    "\n",
    "### 标准状态\n",
    "- 温度：$T_0 = 273 {K}$\n",
    "- 压强：$p_0 = 1.013 \\times 10^5 {Pa}$\n",
    "- 摩尔体积：$V_{m0} = 22.4 {L/mol}$\n",
    "\n",
    "## 3. 气体实验定律\n",
    "\n",
    "### 玻意耳定律（等温变化）\n",
    "$$ p_1V_1 = p_2V_2 \\quad (T {不变}) $$\n",
    "\n",
    "### 查理定律（等容变化）\n",
    "$$ \\frac{p_1}{T_1} = \\frac{p_2}{T_2} \\quad (V {不变}) $$\n",
    "\n",
    "### 盖-吕萨克定律（等压变化）\n",
    "$$ \\frac{V_1}{T_1} = \\frac{V_2}{T_2} \\quad (p {不变}) $$\n",
    "\n",
    "## 4. 热力学第一定律\n",
    "\n",
    "### 热力学第一定律\n",
    "$$ \\Delta U = Q + W $$\n",
    "\n",
    "### 符号规定：\n",
    "- $\\Delta U > 0$：内能增加\n",
    "- $Q > 0$：吸热\n",
    "- $W > 0$：外界对系统做功\n",
    "\n",
    "### 理想气体内能\n",
    "$$ U = \\frac{i}{2}\\nu RT $$\n",
    "其中 $i$ 为自由度（单原子$i=3$，双原子$i=5$）\n",
    "\n",
    "## 5. 功和热量\n",
    "\n",
    "### 气体做功\n",
    "$$ W = p\\Delta V \\quad ({等压过程}) $$\n",
    "$$ W = \\int_{V_1}^{V_2} pdV \\quad ({一般过程}) $$\n",
    "\n",
    "### 热量计算\n",
    "$$ Q = cm\\Delta T \\quad ({不发生相变}) $$\n",
    "$$ Q = \\lambda m \\quad ({熔化/凝固}) $$\n",
    "$$ Q = Lm \\quad ({汽化/液化}) $$\n",
    "\n",
    "## 6. 热机效率\n",
    "\n",
    "### 热机效率\n",
    "$$ \\eta = \\frac{W}{Q_1} = 1 - \\frac{Q_2}{Q_1} $$\n",
    "\n",
    "### 卡诺热机效率\n",
    "$$ \\eta = 1 - \\frac{T_2}{T_1} $$\n",
    "\n",
    "### 制冷系数\n",
    "$$ \\omega = \\frac{Q_2}{W} = \\frac{Q_2}{Q_1 - Q_2} $$\n",
    "\n",
    "## 7. 热传导\n",
    "\n",
    "### 热传导定律\n",
    "$$ \\frac{Q}{t} = \\lambda A \\frac{\\Delta T}{d} $$\n",
    "\n",
    "**其中：**\n",
    "- $\\lambda$：热导率\n",
    "- $A$：横截面积\n",
    "- $d$：厚度\n",
    "- $\\Delta T$：温度差\n",
    "\n",
    "## 8. 物态变化\n",
    "\n",
    "### 熔化热\n",
    "$$ Q = \\lambda m $$\n",
    "\n",
    "### 汽化热\n",
    "$$ Q = Lm $$\n",
    "\n",
    "### 比热容\n",
    "$$ Q = cm\\Delta T $$\n",
    "\n",
    "## 9. 饱和汽\n",
    "\n",
    "### 饱和汽压\n",
    "- 与温度有关，与体积无关\n",
    "- 温度升高，饱和汽压增大\n",
    "\n",
    "### 相对湿度\n",
    "$$ B = \\frac{p}{p_s} \\times 100\\% $$\n",
    "\n",
    "**其中：**\n",
    "- $p$：实际水汽压\n",
    "- $p_s$：该温度下饱和水汽压\n",
    "\n",
    "## 10. 热膨胀\n",
    "\n",
    "### 线膨胀\n",
    "$$ L_t = L_0(1 + \\alpha t) $$\n",
    "$$ \\Delta L = L_0\\alpha\\Delta t $$\n",
    "\n",
    "### 体膨胀\n",
    "$$ V_t = V_0(1 + \\beta t) $$\n",
    "$$ \\Delta V = V_0\\beta\\Delta t $$\n",
    "\n",
    "### 关系\n",
    "$$ \\beta \\approx 3\\alpha $$\n",
    "\n",
    "## 11. 常用常数\n",
    "\n",
    "| 物理量 | 符号 | 数值 |\n",
    "|--------|------|------|\n",
    "| 阿伏伽德罗常数 | $N_A$ | $6.02 \\times 10^{23}$ mol⁻¹ |\n",
    "| 气体常数 | $R$ | $8.31$ J/(mol·K) |\n",
    "| 标准大气压 | $p_0$ | $1.013 \\times 10^5$ Pa |\n",
    "| 冰的熔化热 | $\\lambda$ | $3.35 \\times 10^5$ J/kg |\n",
    "| 水的比热容 | $c$ | $4.18 \\times 10^3$ J/(kg·K) |\n",
    "\n",
    "## 💡 重要概念\n",
    "\n",
    "1. **理想气体**：分子间无相互作用力，分子大小可忽略\n",
    "2. **内能**：与温度有关，与体积无关（理想气体）\n",
    "3. **等值过程**：等温、等容、等压过程的特点\n",
    "4. **热力学第二定律**：热量不能自发地从低温物体传到高温物体\n",
    "\n",
    "## 📝 解题技巧\n",
    "\n",
    "1. 明确研究对象和过程类型\n",
    "2. 注意单位的统一（温度用开尔文）\n",
    "3. 正确判断功的正负号\n",
    "4. 应用能量守恒定律\n",
    "5. 注意理想气体的特殊性质\n",
    "\n",
    "## 12. 图像分析\n",
    "\n",
    "### p-V图\n",
    "- 等温线：双曲线\n",
    "- 等容线：垂直于V轴的直线\n",
    "- 等压线：垂直于p轴的直线\n",
    "\n",
    "### p-T图、V-T图\n",
    "- 等容线：过原点的直线\n",
    "- 等压线：过原点的直线\n",
    "\n",
    "> 掌握这些公式和理解其物理意义是解决热学问题的关键。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90107455",
   "metadata": {},
   "source": [
    "# 单原子分子的绝热过程\n",
    "\n",
    "## 一、什么是绝热过程？\n",
    "\n",
    "在热力学中，**绝热过程** 是指系统与外界 **没有热量交换（Q = 0）** 的过程。也就是说：\n",
    "\n",
    "$$\n",
    "Q = 0\n",
    "$$\n",
    "\n",
    "在这种过程中，如果系统对外做功，其内能会减少；反之，如果外界对系统做功，其内能会增加。\n",
    "\n",
    "对于理想气体，绝热过程满足以下关系（由热力学第一定律与理想气体状态方程推导而来）。\n",
    "\n",
    "---\n",
    "\n",
    "## 二、理想气体的绝热过程（通用情况）\n",
    "\n",
    "对于任意**理想气体（包括单原子分子气体）**，在**绝热过程**中，有以下基本关系式（泊松方程）：\n",
    "\n",
    "### 1. 绝热过程方程（泊松方程）\n",
    "\n",
    "$$\n",
    "PV^\\gamma = \\text{常数}\n",
    "$$\n",
    "\n",
    "$$\n",
    "TV^{\\gamma - 1} = \\text{常数}\n",
    "$$\n",
    "\n",
    "$$\n",
    "P^{1 - \\gamma} T^\\gamma = \\text{常数}\n",
    "$$\n",
    "\n",
    "其中：\n",
    "\n",
    "- $P$：压强  \n",
    "- $V$：体积  \n",
    "- $T$：温度  \n",
    "- $\\gamma = \\dfrac{C_P}{C_V}$：**绝热指数（热容比 / 比热容比）**\n",
    "\n",
    "---\n",
    "\n",
    "## 三、单原子分子气体的特点\n",
    "\n",
    "### 1. 常见的单原子分子气体\n",
    "\n",
    "- 氦（He）\n",
    "- 氖（Ne）\n",
    "- 氩（Ar）\n",
    "- 金属蒸气中的单原子成分等\n",
    "\n",
    "### 2. 自由度与能量\n",
    "\n",
    "- 单原子分子只有 **3 个平动自由度**，没有转动和振动自由度（常温下振动自由度冻结）。\n",
    "- 因此其 **内能只由平动动能贡献**。\n",
    "\n",
    "### 3. 摩尔热容\n",
    "\n",
    "对于**单原子分子理想气体**：\n",
    "\n",
    "- **定容摩尔热容：**\n",
    "  $$\n",
    "  C_V = \\frac{3}{2} R\n",
    "$$\n",
    "\n",
    "- **定压摩尔热容：**\n",
    "  $$\n",
    "  C_P = C_V + R = \\frac{3}{2} R + R = \\frac{5}{2} R\n",
    "$$\n",
    "\n",
    "- **绝热指数（热容比）：**\n",
    "  $$\n",
    "  \\gamma = \\frac{C_P}{C_V} = \\frac{\\frac{5}{2}R}{\\frac{3}{2}R} = \\frac{5}{3} \\approx 1.67\n",
    "$$\n",
    "\n",
    "> ✅ **单原子分子气体的关键参数：$\\gamma = \\frac{5}{3}$**\n",
    "\n",
    "---\n",
    "\n",
    "## 四、单原子分子气体的绝热过程公式\n",
    "\n",
    "由于 $\\gamma = \\frac{5}{3}$，因此对于**单原子分子理想气体**的绝热过程，泊松方程中的 $\\gamma$ 代入为 $\\frac{5}{3}$，得到：\n",
    "\n",
    "### 1. 压强与体积关系\n",
    "$$\n",
    "P V^{5/3} = \\text{常数}\n",
    "$$\n",
    "\n",
    "### 2. 温度与体积关系\n",
    "$$\n",
    "T V^{5/3 - 1} = T V^{2/3} = \\text{常数}\n",
    "$$\n",
    "\n",
    "或者：\n",
    "$$\n",
    "T V^{2/3} = \\text{常数}\n",
    "$$\n",
    "\n",
    "### 3. 压强与温度关系（参考推导）\n",
    "通常由 $P V^\\gamma = \\text{常数}$ 与理想气体状态方程 $P V = n R T$ 联立推导其他组合关系，常见形式为：\n",
    "$$\n",
    "P^{1-\\gamma} T^\\gamma = \\text{常数}\n",
    "\\quad \\text{或} \\quad\n",
    "P T^{-5/2} = \\text{常数} \\quad (\\text{视推导方式而定})\n",
    "$$\n",
    "\n",
    "但最常用的还是从通用的绝热方程和状态方程出发推导两个量之间的关系。\n",
    "\n",
    "---\n",
    "\n",
    "## 五、绝热过程中的功与内能变化\n",
    "\n",
    "### 1. 热力学第一定律（绝热过程）\n",
    "\n",
    "$$\n",
    "\\Delta U = Q - W \\quad \\Rightarrow \\quad \\Delta U = -W \\quad (Q = 0)\n",
    "$$\n",
    "\n",
    "- 系统对外做功（体积增加）：$W > 0$，则 $\\Delta U < 0$，**温度降低**\n",
    "- 外界对系统做功（体积减小）：$W < 0$，则 $\\Delta U > 0$，**温度升高**\n",
    "\n",
    "### 2. 内能（只与温度有关）\n",
    "\n",
    "对于**理想气体（尤其是单原子分子）**，内能变化为：\n",
    "\n",
    "$$\n",
    "\\Delta U = n C_V \\Delta T = n \\cdot \\frac{3}{2} R \\cdot (T_f - T_i)\n",
    "$$\n",
    "\n",
    "---\n",
    "\n",
    "### 3. 绝热过程的功\n",
    "\n",
    "由于 $Q = 0$，有：\n",
    "\n",
    "$$\n",
    "W = -\\Delta U = - n C_V (T_f - T_i) = n C_V (T_i - T_f)\n",
    "$$\n",
    "\n",
    "也可以从 $W = \\int P \\, dV$ 出发，结合 $P V^\\gamma = \\text{常数}$ 积分求出解析式（详见附录）。\n",
    "\n",
    "---\n",
    "\n",
    "## 六、总结：单原子分子绝热过程核心公式\n",
    "\n",
    "| 关系式 | 表达式（单原子分子，$\\gamma = \\frac{5}{3}$） | 说明 |\n",
    "|--------|------------------------------------------------|------|\n",
    "| **绝热过程方程** | $P V^{5/3} = \\text{常数}$ | 压强与体积的关系 |\n",
    "| **温度-体积关系** | $T V^{2/3} = \\text{常数}$ | 温度随体积变化 |\n",
    "| **绝热指数** | $\\gamma = \\frac{C_P}{C_V} = \\frac{5}{3} \\approx 1.67$ | 单原子分子特有 |\n",
    "| **定容热容** | $C_V = \\frac{3}{2} R$ | 仅由平动贡献 |\n",
    "| **定压热容** | $C_P = \\frac{5}{2} R$ | |\n",
    "| **内能变化** | $\\Delta U = n C_V \\Delta T$ | 只与温度变化有关 |\n",
    "| **功与内能关系** | $W = -\\Delta U$ | 绝热过程 |\n",
    "\n",
    "---\n",
    "\n",
    "## 七、举例：绝热压缩/膨胀时的温度变化\n",
    "\n",
    "> **问题：** 一定量的单原子理想气体在绝热过程中体积从 $V_1$ 变为 $V_2$，求温度变化关系。\n",
    "\n",
    "**解：**\n",
    "\n",
    "利用绝热过程温度-体积关系：\n",
    "$$\n",
    "T_1 V_1^{2/3} = T_2 V_2^{2/3}\n",
    "\\Rightarrow T_2 = T_1 \\left( \\frac{V_1}{V_2} \\right)^{2/3}\n",
    "$$\n",
    "\n",
    "- 若 **体积减小（压缩，$V_2 < V_1$）** → $T_2 > T_1$：**温度升高**\n",
    "- 若 **体积增大（膨胀，$V_2 > V_1$）** → $T_2 < T_1$：**温度降低**\n",
    "\n",
    "---\n",
    "\n",
    "## 八、附录：绝热过程的功（推导公式）\n",
    "\n",
    "对于绝热过程，已知 $P V^\\gamma = K$（常数），则气体对外做功为：\n",
    "\n",
    "$$\n",
    "W = \\int_{V_i}^{V_f} P \\, dV = \\int_{V_i}^{V_f} \\frac{K}{V^\\gamma} \\, dV = K \\int_{V_i}^{V_f} V^{-\\gamma} \\, dV\n",
    "$$\n",
    "\n",
    "积分结果为：\n",
    "\n",
    "$$\n",
    "W = \\frac{K}{1 - \\gamma} \\left( V_f^{1 - \\gamma} - V_i^{1 - \\gamma} \\right)\n",
    "= \\frac{P_i V_i^\\gamma}{1 - \\gamma} \\left( V_f^{1 - \\gamma} - V_i^{1 - \\gamma} \\right)\n",
    "$$\n",
    "\n",
    "代入 $\\gamma = \\frac{5}{3}$，可进一步计算具体做功数值（如题目给定初末态）。\n",
    "\n",
    "---\n",
    "\n",
    "## ✅ 总结一句话：\n",
    "\n",
    "> **单原子分子理想气体的绝热过程遵循 $P V^{5/3} = \\text{常数}$，其绝热指数为 $\\gamma = \\frac{5}{3}$，过程中无热量交换，做功直接引起内能与温度的变化。**\n",
    "\n",
    "---\n",
    "\n",
    "如你有具体题目（如求功、温度变化、绝热膨胀后压强等），欢迎提供，我可以继续帮你计算！"
   ]
  }
 ],
 "metadata": {
  "language_info": {
   "name": "python"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}
